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BoolAsSIMDscalar_float_impl hidden_export'a  num_choices"  uniform   uniform_float_implintuniform_int_implSamplerX B1B2 new_inclusive%% sample_single%%  sample_single_inclusive%%    %CHAR_SURROGATE_STARTCHAR_SURROGATE_LENchar_to_comp_u32% %% %%% %% %%%%%      % %%%%%% SampleUniform%UniformSampler% %%%%%&%%&%%%% SampleBorrowBorrowedborrow---- SampleRange&is_empty&. &.impl_sample_range_uweightedweighted_index    % Iupdate_weights""WeightedIndexIter"%/%%%'b %weight"weights" total_weight% WeightZEROchecked_add_assignimpl_weight_intimpl_weight_float           rng   random random_iter random_range random_bool random_ratiofill5 gen_rangegen_bool gen_ratioFill6impl_fill_each6__unsafe impl_fill 6rngs reseeding  Rsdr reseed9next_u32next_u64 fill_bytes999Resultsgenerate9 9 reseed_and_generate99mock 9::smallSeed  from_seedLEN seed_from_u649::xoshiro256plusplus= =   =PHI 9::9::= = THREAD_RNG_RESEED_THRESHOLD94 >9::seq coin_flipper CoinFlipper4chunkchunk_remaining random_ratio_one_over6 flip_c_headsincreasing_uniformIncreasingUniform4nAA  next_index calculate_bound_u32innerRESULT2 IteratorRandomchoose choose_stable choose_multiple_fillchoose_multiple0    IndexedRandom .DEchoose_multiple_array  choose_weightedB% choose_multiple_weighted% IndexedMutRandom choose_mutchoose_weighted_mutG%  SliceRandomshufflepartial_shuffle IRII"  " index_      .into_vec   IntoIter         k  k sample_weighted%sample_efraimidis_spirakis%Elementkey   sample_floyd sample_inplace!zerooneas_usize PPP PPPsample_rejection% sample_array    thread_rng     555566 Bernoullip_int BernoulliErrorInvalidProbability     "Iter4phantom""MapfuncU"K OpenClosed01 KKKKK>KOpen01 KKKKK >K K K K KKKKYYYYY Y Y Y Y YYYYYYz Alphanumeric z z z z z >z Alphabetic zzzzz>zzAz\Gz\Gkz\Gkz \ G k  E  z \ G k  ^   z \ G k  ^  G  z \ G k  ^  _ H  z \ G k  ^  _ _ 0  z \ G k  ^  _ _ 0 J  z \ G k  ^  _ _ 0 a K  z \ G k  ^  _ _ 0 a b L              LOWER_MASK      !           !     !           ! 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Eڂ.7Q"N ! #m.  #.   7% 1m`L\`_6`8J"_( gU ; hE 6 0 - g* '  $ 3Sq4ISitKaSmSySDSDSE$ES,ESST+92g;  Է   # 92gp  | m    C9)0   9 L  Ɛ UƐUKKVƐK)4 Ȑ       T3%` %_Z Y 7    7  M 7s  > 7 %  7      7    $ $V V  V  U%͙U&ސaH pH OBI)H)ߢ  H%W(W P\    X=  pDBӇ>X  46 /   Z , ;R,%%&*'n'n ('n>n)n)n %% %$ZPnD% $nnO o,( (  ))UnTn *7TnTnTn'Snl(U"4"U"%ҏhAR(0lnAo,//&*̃/d/ ׮\,// / //. .׮,.e@},22&*2\2 T 2t2 2 222?$2 2<y,::&*:*d: UL:\::$ : ::9p$9L9\9ן&+T,> > T>>$> $>7> >>>I> $>Bi  i ii iii  i ii ii !.:ȒBIW uȒ %19@N dq} %19@N dq}  %19@N dq}  %19@N dq}# '3;BP hu# '3;BP hu* %19@N dq}* %19@N dq}                                                                                              ,L  CryptoRngL  RngCore< | SeedableRng\  TryCryptoRngd  TryRngCoreT 4 7$ d& | H*' Utilities for random number generationJG Rand provides utilities to generate random numbers, to convert them toKH useful types and distributions, and some randomness-related algorithms. # Quick Start< ```DA // The prelude import enables methods we use below, specificallyPM // Rng::random, Rng::sample, SliceRandom::shuffle and IndexedRandom::choose.̖ use rand::prelude::*; // Get an RNG: let mut rng = rand::rng();FC // Try printing a random unicode code point (probably a bad idea)!1. println!("char: '{}'", rng.random::());85 // Try printing a random alphanumeric value instead!KH println!("alpha: '{}'", rng.sample(rand::distr::Alphanumeric) as char);'$ // Generate and shuffle a sequence: 0- let mut nums: Vec = (1..100).collect();  nums.shuffle(&mut rng); FC // And take a random pick (yes, we didn't need to shuffle first!): " let _ = nums.choose(&mut rng);<  t  # The Book  =: For the user guide and further documentation, please read =: [The Rust Rand Book](https://rust-random.github.io/book).    8l  79https://www.rust-lang.org/logos/rust-logo-128x128-blk.png ;$ 8  7%https://www.rust-lang.org/favicon.ico '$ 8l  7#https://rust-random.github.io/rand/ %    8$  8$  8,  8unused_variables$ 8$  8warningsD <WXWV  M  F | ; / ˒% 4$ 7#" !&   3[ ^ a X M ,    ,   , 8 & 8 *  +   8  8< 7log,  8ԥ'8,   ,   , 8  ,    ,   , 8 & 8 *  +   8  8< 7ԥ,  8ԥ'8,   ,   , 8  $    ,   , 8 & 8 *  +   8  8< 7ԥ,  8ԥ'8$   ,   , 8  $    ,   , 8 & 8 *  +   8  8< 7ԥ,  8ԥ'8$   ,   , 8  ,    ,   , 8 & 8 *  +   8  8< 7ԥ,  8ԥ'8,   ,   , 8  ]`FOR=@(+l<9 Generating random samples from probability distributionsLI This module is the home of the [`Distribution`] trait and several of itsFC implementations. It is the workhorse behind some of the convenientJG functionality of the [`Rng`] trait, e.g. [`Rng::random`] and of course [`Rng::sample`].IF Abstractly, a [probability distribution] describes the probability of1. occurrence of each value in its sample space.NK More concretely, an implementation of `Distribution` for type `X` is anIF algorithm for choosing values from the sample space (a subset of `T`)MJ according to the distribution `X` represents, using an external source of:7 randomness (an RNG supplied to the `sample` function).FC A type `X` may implement `Distribution` for multiple types `T`. IF Any type implementing [`Distribution`] is stateless (i.e. immutable), NK but it may have internal parameters set at construction time (for example, PM [`Uniform`] allows specification of its sample space as a range within `T`).   '$ # The Standard Uniform distribution  MJ The [`StandardUniform`] distribution is important to mention. This is the LI distribution used by [`Rng::random`] and represents the "default" way to MJ produce a random value for many different types, including most primitive LI types, tuples, arrays, and a few derived types. See the documentation of)& [`StandardUniform`] for more details.XU Implementing [`Distribution`] for [`StandardUniform`] for user types `T` makes itMJ possible to generate type `T` with [`Rng::random`], and by extension also! with the [`random`] function.+( ## Other standard uniform distributionsJG [`Alphanumeric`] is a simple distribution to sample random letters andTQ numbers of the `char` type; in contrast [`StandardUniform`] may sample any valid\ `char`.\Y There's also an [`Alphabetic`] distribution which acts similarly to [`Alphanumeric`] but doesn't include digits.NK For floats (`f32`, `f64`), [`StandardUniform`] samples from `[0, 1)`. AlsoHE provided are [`Open01`] (samples from `(0, 1)`) and [`OpenClosed01`]NK (samples from `(0, 1]`). No option is provided to sample from `[0, 1]`; itNK is suggested to use one of the above half-open ranges since the failure toKH sample a value which would have a low chance of being sampled anyway is  rarely an issue in practice.)& # Parameterized Uniform distributionsKH The [`Uniform`] distribution provides uniform sampling over a specifiedHE range on a subset of the types supported by the above distributions.52 Implementations support single-value-sampling via41 [`Rng::random_range(Range)`](Rng::random_range).NK Where a fixed (non-`const`) range will be sampled many times, it is likely;8 faster to pre-construct a [`Distribution`] object usingB? [`Uniform::new`], [`Uniform::new_inclusive`] or `From`.ԕ # Non-uniform samplingMJ Sampling a simple true/false outcome with a given probability has a name:JG the [`Bernoulli`] distribution (this is used by [`Rng::random_bool`]).IF For weighted sampling of discrete values see the [`weighted`] module.JG This crate no longer includes other non-uniform distributions; insteadGD it is recommended that you use either [`rand_distr`] or [`statrs`].VS [probability distribution]: https://en.wikipedia.org/wiki/Probability_distribution74 [`rand_distr`]: https://crates.io/crates/rand_distr/, [`statrs`]: https://crates.io/crates/statrs [`random`]: crate::random7/,"KYzSL"Tt"d"F d#&U$#U#W4#WVd#V[T#[[d$[}<$ $l L  d&]ԘAT1L \,$`8  ``` ]`]9ֹ`  $ )  8< 7,t7JG Construct a new `Bernoulli` with the given probability of success `p`.| # PrecisionHE For `p = 1.0`, the resulting distribution will always generate true.IF For `p = 0.0`, the resulting distribution will always generate false.LI This method is accurate for any input `p` in the range `[0, 1]` which isB? a multiple of 2-64. (Note that not all multiples of?< 2-64 in `[0, 1]` can be represented as a `f64`.) /%X!B? Construct a new `Bernoulli` with the probability of success of"DA `numerator`-in-`denominator`. I.e. `new_ratio(2, 3)` will return"JG a `Bernoulli` with a 2-in-3 chance, or about 67%, of returning `true`.##EB return `true`. If `numerator == 0` it will always return `false`.$IF For `numerator > denominator` and `denominator == 0`, this returns an$HE error. Otherwise, for `numerator == denominator`, samples are always%85 true; for `numerator == 0` samples are always false.T% ן~*)63 Returns the probability (`p`) of the distribution.**KH This value may differ slightly from the input due to loss of precision. *  * $*+% ( ,64,  !! , ,, $,4*, d &F"#4H"$ $2"%T>-T ́ NK Types (distributions) that can be used to create a random instance of `T`.A> It is possible to sample from a distribution through both theGD `Distribution` and [`Rng`] traits, via `distr.sample(&mut rng)` andOL `rng.sample(distr)`. They also both offer the [`sample_iter`] method, which<9 produces an iterator that samples from the distribution.NK All implementations are expected to be immutable; this has the significantDA advantage of not needing to consider thread safety, and for mostIF distributions efficient state-less sampling algorithms are available.KH Implementations are typically expected to be portable with reproducible :7 results when used with a PRNG with fixed seed; see the NK [portability chapter](https://rust-random.github.io/book/portability.html) NK of The Rust Rand Book. In some cases this does not apply, e.g. the `usize` C@ type requires different sampling on 32-bit and 64-bit machines.  .+ [`sample_iter`]: Distribution::sample_iterd &''&  &*0(*,(*,,*(24 4 LI Generate a random value of `T`, using `rng` as the source of randomness.4   &)) &   ($ 4 35b JG Create an iterator that generates random values of `T`, using `rng` as  the source of randomness.C@ Note that this function takes `self` by value. This works sinceDA `Distribution` is impl'd for `&D` where `D: Distribution`,IF however borrowing is not automatic hence `distr.sample_iter(...)` mayEB need to be replaced with `(&distr).sample_iter(...)` to borrow orC@ `(&*distr).sample_iter(...)` to reborrow an existing reference.l # Example<LI use rand::distr::{Distribution, Alphanumeric, Uniform, StandardUniform}; // Vec of 16 x f32:OL let v: Vec = StandardUniform.sample_iter(&mut rng).take(16).collect();t // String:  let s: String = Alphanumeric .sample_iter(&mut rng) .take(7)İ .map(char::from) .collect(); // Dice-rolling::7 let die_range = Uniform::new_inclusive(1, 6).unwrap();74 let mut roll_die = die_range.sample_iter(&mut rng);)& while roll_die.next().unwrap() != 6 {,) println!("Not a 6; rolling again!");, }<\&++& , *4:<g" Map sampled values to type `S`l<-* use rand::distr::{Distribution, Uniform};41 let die = Uniform::new_inclusive(1, 6).unwrap();2/ let even_number = die.map(|num| num % 2 == 0);1. while !even_number.sample(&mut rand::rng()) {.+ println!("Still odd; rolling again!");,<©&-.-.&© Я  ©T©  Я , ,©Я Vkmgi;80110  &|2(2460234  /33 /  28$4̕79"R567567Ѯ "׮ "ݮ "&Ѯݮ|"׮"89:KKK9:egac]_L"$"44ݮ#$# ۲4 #4 9Ѯ׮ݮ$#%-L% 4 %4 :Ѯ׮ݮ$&&]<=>=>< &ȇ &ه & |' '(*-/24)e@ABCB@ACƯ Я *Ư )ه ) ) |* ƯT* )D(DQSVX[]*34*  Я?EE *? ** DѮƯݮЯ$*4k:<,+! Sample or extend a [`String`]++EB Helper methods to extend a [`String`] or sample a new [`String`]., , ,8<, ,7<,̓,d,FF,FGIGIIG.W,)& Append `len` random chars to `string`,-JG Note: implementations may leave `string` with excess capacity. If this-IF is undesirable, consider calling [`String::shrink_to_fit`] after this\. method.l.  ׮ FHH .F׮ . .$.4.4. ./11K//, Generate a [`String`] of `len` random chars//LI Note: implementations may leave the string with excess capacity. 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N K [`Uniform`] is the standard distribution to sample uniformly from a range;X U e.g. `Uniform::new_inclusive(1, 6).unwrap()` can sample integers from 1 to 6, like aH E standard die. [`Rng::random_range`] is implemented over [`Uniform`].   # Example usage <   use rand::Rng;  use rand::distr::Uniform;  2 / let side = Uniform::new(-10.0, 10.0).unwrap(); % " // sample between 1 and 10 points* ' for _ in 0..rng.random_range(1..=10) {O L // sample a point from the square with sides -10 - 10 in two dimensions: 7 let (x, y) = (rng.sample(side), rng.sample(side));( % println!("Point: {}, {}", x, y);, <   2 / # Extending `Uniform` to support a custom type  K H To extend [`Uniform`] to support your own types, write a back-end which Q N implements the [`UniformSampler`] trait, then implement the [`SampleUniform`] L I helper trait to "register" your back-end. See the `MyF32` example below.  P M At a minimum, the back-end needs to store any parameters needed for sampling N K (e.g. the target range) and implement `new`, `new_inclusive` and `sample`. O L Those methods should include an assertion to check the range is valid (i.e. C @ `low < high`). The example below merely wraps another back-end.  M J The `new`, `new_inclusive`, `sample_single` and `sample_single_inclusive`  functions use arguments ofG D type `SampleBorrow` to support passing in values by reference orI F by value. In the implementation of these functions, you can choose toT Q simply use the reference returned by [`SampleBorrow::borrow`], or you can chooseF C to copy or clone the value, whatever is appropriate for your type. < ̜ 6 3 use rand::distr::uniform::{Uniform, SampleUniform,? < UniformSampler, UniformFloat, SampleBorrow, Error};   struct MyF32(f32); !  #[derive(Clone, Copy, Debug)]+ ( struct UniformMyF32(UniformFloat); * ' impl UniformSampler for UniformMyF32 {  type X = MyF32; @ = fn new(low: B1, high: B2) -> Result4 1 where B1: SampleBorrow + Sized,3 0 B2: SampleBorrow + SizedL  {W T UniformFloat::::new(low.borrow().0, high.borrow().0).map(UniformMyF32)L  }J G fn new_inclusive(low: B1, high: B2) -> Result4 !3 !L !a ^ UniformFloat::::new_inclusive(low.borrow().0, high.borrow().0).map(UniformMyF32)L !C @ fn sample(&self, rng: &mut R) -> Self::X {% " MyF32(self.0.sample(rng))L !,  "  impl SampleUniform for MyF32 {$ ! type Sampler = UniformMyF32;,  7 4 let (low, high) = (MyF32(17.0f32), MyF32(22.0f32));3 0 let uniform = Uniform::new(low, high).unwrap();- * let x = uniform.sample(&mut rand::rng());<  ; 8 [`SampleUniform`]: crate::distr::uniform::SampleUniform= : [`UniformSampler`]: crate::distr::uniform::UniformSampler5 2 [`UniformInt`]: crate::distr::uniform::UniformInt9 6 [`UniformFloat`]: crate::distr::uniform::UniformFloat? < [`UniformDuration`]: crate::distr::uniform::UniformDurationI F [`SampleBorrow::borrow`]: crate::distr::uniform::SampleBorrow::borrow to ensure that samples from `[low, high)` are always strictlyA<7 less than `high`, this approach may be very slow where:<0 `scale.abs()` is much smaller than `high.abs()`3</ (example: `low=0.99999999997819644, high=1.`).28 8h\  8g& , 8$ 8$& , 8$ 88g,& , 8(8$  88D  , 8 '8!,   7 8, 8ן D , 8ן D'8 <% 8$  88$   8g, 8D 8g 8 <  8$% 8 8$ 8    8,% 8g, 8g, 8 |  8$% 8!d     , 8 '8!,  70.0 8 4  8g,% 8gd  8g$ 8g,,   +   8  , 8 $- 8$8+l8 , 8   8!$8%< 8gd , 8 % ,   +   8  , 8 $- 8$8+t8 8gd , 8   <8!$8%  , 8% 8 8  8%$ 8%   8,& 8%$ 84& 8%(84 8$$ 8, 8$,8%& 8-d 8$'8%   8,$ 8%& 8-d 8$'8%   8,$   !,88g  8, 8-4  % 88$  84 8-4  % +   8  88   8g 8 T     8$ 8 T    848  8,'8|L% 8   8g 8 4  8$  848  8,'8|T% 8 8g,  8$  8g % 8    8g,  8 T     !84 8   8, ' 8|L % 8! ! !8$!'!8h\! ! !8g!$ !8$!$ !8g,!8 !8%l! !8%!$ !8%! ! ! "8,!& !8%!$ !84!& "8%"("84" "8$"$ "8," "8$,"8%"& "8-d" "8$"'"8% " " "8,"$ "8%"& "8-d" #8$#'#8% # # #8,#$ # # (=8#8g# # #8,# #8-4# # #% #8#8$# # #84# #8-4# # #% #+ $ $ $8$ $ $8$8$ $ $ $8g$ $8 T$ $ $$ $ $ $8$$ $8 T$ $ $ $ %84$8% % %8,%'%8|L%% %8% %8g% %8 4% % %8$% % &84%8% % &8,%'%8|T%% &8&8D& & &, &8& &'&8!,& & & 7 &8&, &8ן D& &, &8ן D&'&8 <&% &8&8g,' ' ' '8$' '8g' '8D'% '8' '8g,' '8 T' ' ' ' (84'8' ' '8,'''8|L'% '8( ( (8$('(8h\( ( (8g($ (8$($ (8g,(8 (84( (8 (& (8( (- (8,( ( ( ) )8$)$ )84)& ) )8)8 )()8$)')8% ) ) /8)8D) ) ) *84) )854)'* *, *8 * * * **, *8 *'*8!,* * *, *8"|* *8İ* * *7 *% *8+8D+ +8D+ + +, +8+ +'+8!,+ + +7 +% +8D/ /8$/ /8g,/ /8$/ /8g/+ / / /84/8 /8&l/ /8 /& /8/ /- /8,/$ /8%/$ /8%/ / / 0 8,/& 08%0$ 0840& 08%0$ 0840& 0 0808 0(0840 08$0'08% 0$ 08,0 08$,08%0& 08-d0 08$0'08% 0 0 08,0$ 18%1& 18-d1 18$1'18% 1 1 18,1$ 1 1 28$1'18&1 1 28,1$ 1841$ 1842+ 2 2 28428 28&2 28 2& 282 2- 28,2$ 28%2$ 28%2 2 2 3 8,2& 28%2$ 2842& 28%2$ 2843& 3 3838 3(3843 38$3'38% 3$ 38,3 38$,38%3& 38-d3 38$3'38% 3 3 38,3$ 38%3& 38-d4 48$4'48% 4 4 48,4$ 4 4 <M848g4 4 48,4 48-44 4 4% 4848$4 4 4844 48-44 5 5% 5+ 5 5 585 5 58585 58g5 58 T5 5 55 58$5 58 T5 5 5 5 684686 6 68,6'68|L6% 686 68g6 68 46 6 68$6 6 784686 6 78,6'68|T6% 7878g,7 78$7 78g7% 787 78g,7 78 T7 7 7 7 884888 8 88,8'88|L8% 8888D8 9 9 9849 98549'9 9, 98 9 9 9 99, 98 9'98!,9 9 9, 98"|9 989 9 97 9% 98:8D: :8D: : :, :8: ;';8!,; ; ;7 ;% ;8< < <8D< <8g,< <8g<% < p#<   , -d+l+t;subdQS%ET6  R  , 8& 8$ , 8 & 8$ , 8 sample_tyL& 8,*  R8$8+l8 , 8   8!$8%< 8jT , 8 % 8$8+t8 8jT , 8   R8!$8%  , 8% +   848 8  8%$ 8%   8,& 8%$ 84& 8%(84 8$$ 8, 8$,8%& 8-d 8$'8%   8, $ 8% & 8-d  8$ ' 8%   8, $   "8 8g   8,  8-4   % 8!8$! ! !84! !8-4! ! !% !8! ! ! !8g! !8$! ! "84!8! ! !8,!'!8|T!% !8+t"'"8%l" " "8g"$ "8$" "7 "+ " " "84"8 #8%l# #8%#$ #8%# # # $8,#& $8%$$ $84$& $8%$($84$ $8$$$ $8,$ $8$,$8%$& $8-d$ $8$$'$8% $ $ $8,$$ $8%%& %8-d% %8$%'%8% % % %8,%$ % % *18%8g% % %8,% %8-4% % %% %8%8$% % %84% %8-4& & &% &8& & & &8g&&8$& & &84&8& & &8,&'&8|T&% &8'8e,' '8$' '8d' ' '8g' '8d' ' '7 '8', '8 '% '8'8k4' '8'8e,' '7 ' ' ( 8(8e,( (, (8"L('(8$( ( (8e,(% ( ( (8e,( (8 wrapping_negd( ( ( (8e,(8$( ( )7 )% )8) ) *8jT) ) *8g)$ )8e,)& )8e,)8), )8)$ )8k4*& *8k4*8*, *8 *8*, *8*$ *<4 Sample from distribution, Lemire's method, unbiased+7+ + + +84+8 +84+ +8 +& +8+ +- +8,+ + + , +8$,$ ,84,& , ,8,8 ,(,8$,',8% , , 0+8,8e,, ,8$, ,8e,,8,, ,8 ,8,, ,8"L,% ,8,8e,,,7 , , -84-84- -854- - -% -8-8k4- -8$- -8k4-8-, -8 -8-, -8"L-% -8.8. .8$. . /8. . .8.$ .8. .84. .854.'. ., .8"L. . . . .8$. . .8e,.% .8.8..8k4/ / /8,/8/% /% /8$/ /8g/ /8d/ / 08/8/, 080+ 0 0 08408 08&l0 08 0& 080 0- 08,0$ 08%0$ 08%0 0 0 18,0& 08%0$ 0841& 18%1$ 1841& 1 1818 1$ 1(1841 18$1'18% 1$ 18,1 18$,18%2& 28-d2 28$2'28% 2 2 28,2$ 28%2& 28-d2 28$2'28% 2 2 28,2$ 2 2 4828g2 2 28,3 38-43 3 3% 3838$3 3 3843 38-43 3 3% 383 3 3 38g3 38$3 3 484383 3 48,3'48|T4% 48$4'48&4 4 48g4$ 48$4 47 4$ 4844<, Sample single value, Canon's method, biased5/<5<> In the worst case, bias affects 1 in `2^n` samples where n is5A<= 56 (`i8`), 48 (`i16`), 96 (`i32`), 64 (`i64`), 128 (`i128`).6@+ 6 6 786 6 78 6 6 78<6 67unbiasedT6+ 7 7 78478 78&7 78 7& 787 7- 78,7$ 78%7$ 78%7 7 7 88,7& 78%7$ 7848& 88%8$ 8848& 8 8888 8$ 8(8848 88$8'88% 8$ 88,8 88$,88%9& 98-d9 98$9'98% 9 9 98,9$ 98%9& 98-d9 98$9'98% 9 9 98,9$ 9 9 BF898g9 9 98,9 :8-4: : :% :8:8$: : :84: :8-4: : :% :8: : : :8g::8$: : ;84:8: : ;8,:';8|T;% ;8;8e,; ;8$; ;8d; ; ;8g; ;8d; ; ;7 ;8;, ;8 ;8;, ;8"L;% ;8<8e,<<7 < < =84=8= = =84= =854= = =% =8> > >8>8 4>$ >8D> >84> >854>'> >, >8"L> > > > >8$> > >8e,>% >8?8D? ?8e,? ?8"d? ? ? ? B8@ @ @8d@$ @8 @ @ @ @ 84@ @854@'@ @, @8"L@ @ @ @ @8$@ @ @8e,@8@, @8"L@% @8A8\A A8DA A8 checked_add\A A A8dA8A, A8"LA A8is_none= high`, or if `low`, `high` or the range `high - low` isF; 8 non-finite. In release mode, only the range is checked.F  %%#,G |G #,H |G gF F J HL I Create a new `Uniform` instance, which samples uniformly from the closedI$ ! range `[low, high]` (inclusive).I IK H Fails if `low > high`, or if `low`, `high` or the range `high - low` isJ; #lJ  %%#,K #|K #,K #|K gJJJ J L5 %# L һ#lL (M3 4M   ׮ M  M M ׮$M 4M')O MI F Helper trait for creating objects using the correct implementation ofN- * [`UniformSampler`] for the sampling type.N NH E See the [module documentation] on how to implement [`Uniform`] rangeO  sampling for a custom type.O O1 . [module documentation]: crate::distr::uniformlO ,P P' #P #DP #,#!#P& P< 9 The `UniformSampler` implementation supporting type `X`.

Usually users should not call this directly but prefer to useX  [`Uniform::new`].X  %%#,Y  Y #,Y  Y gX $Y X X [ Z8 5 Construct self, with inclusive bounds `[low, high]`.Z ZA #[  [`Uniform::new_inclusive`].l[  %%#,\ #\ #,\ #\ g[ $[ [ [ \: \  Sample a value.4\   ׮  ]  ] ] $] 4]  f ]K H Sample a single value uniformly from a range with inclusive lower bound^, ) and exclusive upper bound `[low, high)`.^ ^L #_N #_D ˽#ĭ` #` `( % By default this is implemented using`I F `UniformSampler::new(low, high).sample(rng)`. However, for some typesaJ G more optimal implementations for single usage may be provided via thisb7 4 method (which is the case for integers and floats).b!  Results may not be identical.b cK H Note that to use this method in a generic context, the type needs to bec6 3 retrieved via `SampleUniform::Sampler` as follows: ; use rand::distr::uniform::{SampleUniform, UniformSampler};d  # #[allow(unused)]d? < fn sample_from_range(lb: T, ub: T) -> T {e"  let mut rng = rand::rng();eO L ::Sampler::sample_single(lb, ub, &mut rng).unwrap(),f  Immutably borrows from an owned value. See [`Borrow::borrow`]t t3 0 [`Borrow::borrow`]: std::borrow::Borrow::borrow4u    u  $u uU -Du lv #%v 4v  # v  $v vV 8-#Dw lw #%w 4w    w  8$w y x? < Range that supports generating a single sample efficiently.x xM J Any type implementing this trait can be used to specify the sampled rangey  for `Rng::random_range`.\y y  y "(35zM y+ ( Generate a sample from the given range.lz   z  z |`] exists.  Error cases:E B - [`Error::InvalidInput`] when the iterator `weights` is empty.K H - [`Error::InvalidWeight`] when a weight is not-a-number or negative.K H - [`Error::InsufficientNonZero`] when the sum of all weights is zero. B ? - [`Error::Overflow`] when the sum of all weights overflows.  2 / [`Uniform`]: crate::distr::uniform::Uniform! 0 ! Jd" |" 4" 2ݦMO4 +G D Update a subset of weights, without changing the number of weights., ,. + `new_weights` must be sorted by the index., ,X U Using this method instead of `new` might be more efficient if only a small number of-V S weights is modified. No allocations are performed, unless the weight type `X` usesԀ.  allocation internally.. .: 7 In case of error, `self` is not modified. Error cases:.E B - [`Error::InvalidInput`] when `new_weights` are not ordered by/' $ index or an index is too large./K $0K ̖$0K H Note that due to floating-point loss of precision, this case is not1L I always correctly detected; usage of a fixed-point weight type may be2  preferred.2 2P M Updates take `O(N)` time. If you need to frequently update weights, consider3c ` [`rand_distr::weighted_tree`](https://docs.rs/rand_distr/*/rand_distr/weighted_tree/index.html)34 1 as an alternative where an update is `O(log N)`.t4      4  5#  5# ,5 <6 $4  5 5 K? K "% K lK TK "K$ 0/b&0XJ"b&0XJK rtK  "dK ,K Lr 8%ڥ$ L $lL $TL `,L `,L `JLL8 M   ``` ]`]9ֹ` M  8$M NS 8%ڥ$ N $lO $TO #%O ,O   O  8$O P 8%ڥ$ P  Q# $lQ $TQ ,Q KK_aQ LQ $Q Q( $Q  92gK R  8$R S< %# S һ#lS $TT פ$,T )+Xi T8 5 Returns the weight at the given index, if it exists.T T; 8 If the index is out of bounds, this will return `None`.U lU U >(), vec![1, 2, 3]);`- * dist.update_weights(&[(0, &2)]).unwrap();`B ? assert_eq!(dist.weights().collect::>(), vec![2, 2, 3]);(rng: &mut R)`. Some!  things are worth noting here: L I - Since `Rng: RngCore` and every `RngCore` implements `Rng`, it makes no 9 6 difference whether we use `R: Rng` or `R: RngCore`. J G - The `+ ?Sized` un-bounding allows functions to be called directly on P M type-erased references; i.e. `foo(r)` where `r: &mut dyn RngCore`. Without 8 5 this it would be necessary to write `foo(&mut r)`.  O L An alternative pattern is possible: `fn foo(rng: R)`. This has some O L trade-offs. It allows the argument to be consumed directly without a `&mut` N K (which is how `from_rng(rand::rng())` works); also it still works directly N K on references (including type-erased references). Unfortunately within the L I function `foo` it is not known whether `rng` is a reference type or not,J G hence many uses of `rng` require an extra reference, either explicitlyO L (`distr.sample(&mut rng)`) or implicitly (`rng.random()`); one may hope the4 1 optimiser can remove redundant references later. d Example: <  ! 1 . fn foo(rng: &mut R) -> f32 {  rng.random(),  $ ! # let v = foo(&mut rand::rng());<  F |< )$5$* P C @ Return a random value via the [`StandardUniform`] distribution. l  <  !    let x: u32 = rng.random();  println!("{}", x);2 / println!("{:?}", rng.random::<(f64, bool)>());<    # Arrays and tuples 8 5 The `rng.random()` method is able to generate arraysG D and tuples (up to 12 elements), so long as all element types can bet generated. I F For arrays of integers, especially for those with small element typesF C (< 64 bit), it will likely be faster to instead use [`Rng::fill`],2 / though note that generated values will differ. <  !  I F let tuple: (u8, i32, char) = rng.random(); // arbitrary tuple support D A let arr1: [f32; 32] = rng.random(); // array construction  let mut arr2 = [0u8; 128];9 6 rng.fill(&mut arr2); // array fill<  / , [`StandardUniform`]: distr::StandardUniform4      &| $ !#  = : Return an iterator over [`random`](Self::random) variates < 9 This is a just a wrapper over [`Rng::sample_iter`] using  [`distr::StandardUniform`]. 0 - Note: this method consumes its argument. Use: 7 `(&mut rng).random_iter()` to avoid consuming the RNG. l  < 1 . use rand::{rngs::SmallRng, Rng, SeedableRng}; ) & let rng = SmallRng::seed_from_u64(0);: 7 let v: Vec = rng.random_iter().take(5).collect();  assert_eq!(v.len(), 5);< \ Ι ,! $|! L(**n !/ , Generate a random value in the given range." "H E This function is optimised for the case that only a single sample is"D A made from the given range. See also the [`Uniform`] distribution#H E type which may be faster if sampling from the same range repeatedly.$ $J G All types support `low..high_exclusive` and `low..=high` range syntax.$P M Unsigned integer types also support `..high_exclusive` and `..=high` syntax.% d% # Panics% %J G Panics if the range is empty, or if `high - low` overflows for floats.& l& & <& & !& & ' '  // Exclusive range'( % let n: u32 = rng.random_range(..10);'  println!("{}", n);(0 - let m: f64 = rng.random_range(-40.0..1.3e5);(  println!("{}", m);( (  // Inclusive range() & let n: u32 = rng.random_range(..=10);) %<) ) )( % [`Uniform`]: distr::uniform::UniformR|* d*   *  *  * l* t* $* ePBDGI0) ,7 4 Return a bool with a probability `p` of being true., ,C @ See also the [`Bernoulli`] distribution, which may be faster if-2 / sampling from the same probability repeatedly.- l- - <- - !. . ./ , println!("{}", rng.random_bool(1.0 / 3.0));<. . d/ %/ ԛ/  If `p < 0` or `p > 1`./ /# [`Bernoulli`]: distr::BernoulliR|/ \0   0 $0 X8D 1H E Return a bool with a probability of `numerator/denominator` of beingL2  true.2 2H E That is, `random_ratio(2, 3)` has chance of 2 in 3, or about 67%, of3J G returning true. If `numerator == denominator`, then the returned value3F C is guaranteed to be `true`. If `numerator == 0`, then the returned4& # value is guaranteed to be `false`.4 4C %5D A sampling from the same `numerator` and `denominator` repeatedly.5 d5 %6 67 4 If `denominator == 0` or `numerator > denominator`.6 l6 6 <6 6 !7 7 7+ ( println!("{}", rng.random_ratio(2, 3));<7 7 8# %R|8 d8   8 $8 ן_bhb>: ;5 2 Sample a new value, using the given distribution.; |; ### Example; <; < !< !< < <9 6 let x = rng.sample(Uniform::new(10u32, 15).unwrap());=I F // Type annotation requires two types, the type and distribution; the=$ ! // distribution can be inferred.>@ = let y = rng.sample::(Uniform::new(10, 15).unwrap());<> 4>   ?  >  > |? $? 9f=?9;Hz ?J G Create an iterator that generates values using the given distribution.@ @1 . Note: this method consumes its arguments. Use@< 9 `(&mut rng).sample_iter(..)` to avoid consuming the RNG.A lA A  ; use rand::distr::{Alphanumeric, Uniform, StandardUniform};B B B B BQ N let v: Vec = (&mut rng).sample_iter(StandardUniform).take(16).collect();C tC C8 5 let s: String = (&mut rng).sample_iter(Alphanumeric)D ĻD D D D  // Combined valuesED A println!("{:?}", (&mut rng).sample_iter(StandardUniform).take(5)EA > .collect::>());F F F: G9 6 let mut roll_die = (&mut rng).sample_iter(die_range);G) G, ,H ACO2 I8 5 Fill any type implementing [`Fill`] with random dataJ JJ G This method is implemented for types which may be safely reinterpretedJJ G as an (aligned) `[u8]` slice then filled with random data. It is oftenK? < faster than using [`Rng::random`] but not value-equivalent.L LI F The distribution is expected to be uniform with portable results, butL> ; this cannot be guaranteed for third-party implementations.M lM M ACU& T# Alias for [`Rng::random_bool`].%Renamed to `random_bool`TA DU %%  U  $U iWA V$ ! Alias for [`Rng::random_ratio`].%Renamed to `random_ratio`VB LW %%  W  $W ןUj^jX#  X |; See also [Types of generators] and [Our RNGs] in the book. ## GeneratorsUR This crate provides a small selection of non-[portable] random number generators:TQ - [`OsRng`] is a stateless interface over the operating system's random numberOL source. This is typically secure with some form of periodic re-seeding.EB - [`ThreadRng`], provided by [`crate::rng()`], is a handle to aQN thread-local generator with periodic seeding from [`OsRng`]. Because thisJG is local, it is typically much faster than [`OsRng`]. It should be74 secure, but see documentation on [`ThreadRng`].PM - [`StdRng`] is a CSPRNG chosen for good performance and trust of security RO (based on reviews, maturity and usage). The current algorithm is ChaCha12, :7 which is well established and rigorously analysed. KH [`StdRng`] is the deterministic generator used by [`ThreadRng`] but B? without the periodic reseeding or thread-local management. NK - [`SmallRng`] is a relatively simple, insecure generator designed to be FC fast, use little memory, and pass various statistical tests of  randomness quality.  MJ The algorithms selected for [`StdRng`] and [`SmallRng`] may change in any A> release and may be platform-dependent, therefore they are not [reproducible][portable]. ### Additional generatorsKH - The [`rdrand`] crate provides an interface to the RDRAND and RDSEED<9 instructions available in modern Intel and AMD CPUs.IF - The [`rand_jitter`] crate provides a user-space implementation ofJG entropy harvesting from CPU timer jitter, but is very slow and hasJG [security issues](https://github.com/rust-random/rand/issues/699).HE - The [`rand_chacha`] crate provides [portable] implementations ofEB generators derived from the [ChaCha] family of stream ciphersNK - The [`rand_pcg`] crate provides [portable] implementations of a subset96 of the [PCG] family of small, insecure generatorsMJ - The [`rand_xoshiro`] crate provides [portable] implementations of the63 [xoshiro] family of small, insecure generators1. For more, search [crates with the `rng` tag]. ## Traits and functionalityKH All generators implement [`RngCore`] and thus also [`Rng`][crate::Rng].52 See also the [Random Values] chapter in the book.C@ Secure RNGs may additionally implement the [`CryptoRng`] trait.@= Use the [`rand_core`] crate when implementing your own RNGs.DA [portable]: https://rust-random.github.io/book/crate-reprod.htmlLI [Types of generators]: https://rust-random.github.io/book/guide-gen.htmlB? [Our RNGs]: https://rust-random.github.io/book/guide-rngs.htmlIF [Random Values]: https://rust-random.github.io/book/guide-values.html [`Rng`]: crate::Rng [`RngCore`]: crate::RngCore# [`CryptoRng`]: crate::CryptoRng'$ [`SeedableRng`]: crate::SeedableRng/, [`rdrand`]: https://crates.io/crates/rdrand96 [`rand_jitter`]: https://crates.io/crates/rand_jitter96 [`rand_chacha`]: https://crates.io/crates/rand_chacha30 [`rand_pcg`]: https://crates.io/crates/rand_pcg;8 [`rand_xoshiro`]: https://crates.io/crates/rand_xoshiro?< [crates with the `rng` tag]: https://crates.io/keywords/rng*' [chacha]: https://cr.yp.to/chacha.html&# [PCG]: https://www.pcg-random.org/(% [xoshiro]: https://prng.di.unimi.it/$dҌD#4#L$,$  lL\ ґD   BlockRngCored  CryptoBlockRngt  L  <  |\  d  ˒T   : F subdPR>@b 9  $ d \ T NPJLG J G Create a new `ReseedingRng` from an existing PRNG, combined with a RNG  to use as reseeder. L I `threshold` sets the number of generated bytes after which to reseed theI F PRNG. Set it to zero to never reseed based on the number of generated\  values.  3 $ ! Immediately reseed the generator 9 6 This discards any remaining random data in the cache.4     $ z 9 T %$ %IJ %  %\ %T  } ~ 68SU! D!   !  $! ! D!   !  $! ") T"    "  $" #| 9% # %$# %d# %\# ,# %T# ,# SUXZ$( ,$   $  $$ % 9%T& %$% %Ē& % % %\& t& d& /1LN't 9% ' %$' %d( %\( %T(    >@CEL( $( d) <) )3 D)     )  $) {-c 9% - %$- %d. %\. %T. 1368/G / ֟ǕDD0 5/ 45   5 \D6    7P 7   & 7 $7 :} 9% : %$: %d; %\; &,; %T; &,; CEHJ;) ,;   ;  $; = 9%T> %$> %> % > %\> &t> &d?  "=? d  Mock random number generator0.9.2$&impls, G< |ܡ*,d &0 EB Create a `StepRng`, yielding an arithmetic sequence starting with 74 `initial` and incremented by `increment` each time.& rrĠ& } ~ D&   $D&   $(T&    $L    8<  7 small_rng\  , < |\ !,.D  8 7ʓ#$#   L$&L  $l P̽  } ~  D    $!!D!  ! $!")T"   " $"!! ! !8! ! !8());,B? println!("A simulated die roll: {}", rng.random_range(1..=6));<- # }<--t- # Security--$! Refer to [`ThreadRng#Security`]...    .̂/+-T4 T,|%% %<" 4C %T       + 3#,ĵ<dD A\FlA ^OT<|)*DA Extension trait on iterators, providing random sampling methods.NK This trait is implemented on all iterators `I` where `I: Iterator + Sized` and provides methods for=: choosing one or more elements. You must `use` this trait:<" use rand::seq::IteratorRandom;+( let faces = "😀😎😐😕😠😢";JG println!("I am {}!", faces.chars().choose(&mut rand::rng()).unwrap());<'$ Example output (non-deterministic):\ ```nonet I am 😀!<t,KD(&4&*\  Uniformly sample one elementEB Assuming that the [`Iterator::size_hint`] is correct, this method HE returns one uniformly-sampled random element of the slice, or `None` GD only if the slice is empty. Incorrect bounds on the `size_hint` may IF cause this method to incorrectly return `None` if fewer elements than HE the advertised `lower` bound are present and may prevent sampling of KH elements beyond an advertised `upper` bound (i.e. incorrect `size_hint` B? is memory-safe, but may result in unexpected `None` result and  non-uniform distribution).  KH With an accurate [`Iterator::size_hint`] and where [`Iterator::nth`] is HE a constant-time operation, this method can offer `O(1)` performance.̻ Where no size hint isEB available, complexity is `O(n)` where `n` is the iterator length.?< Partial hints (where `lower > 0`) also improve performance.JG Note further that [`Iterator::size_hint`] may affect the number of RNGJG samples used as well as the result (while remaining uniform sampling).EB Consider instead using [`IteratorRandom::choose_stable`] to avoidLI [`Iterator`] combinators which only change size hints from affecting thed results.l<"&41 let words = "Mary had a little lamb".split(' ');<9 println!("{}", words.choose(&mut rand::rng()).unwrap());<4 ׮    ׮$4[x(*,c!)& Uniformly sample one element (stable)""DA This method is very similar to [`choose`] except that the result"IF only depends on the length of the iterator and the values produced by#HE `rng`. Notably for any iterator of a given length this will make the#JG same requests to `rng` and if the same sequence of values are produced$JG the same index will be selected from `self`. This may be useful if you%KH need consistent results no matter what type of iterator you are working%>; with. If you do not need this stability prefer [`choose`].&&DA Note that this method still uses [`Iterator::size_hint`] to skip&IF constructing elements where possible, however the selection and `rng`'GD calls are the same in the face of this optimization. If you want to(IF force every element to be created regardless call `.inspect(|e| ())`.((&# [`choose`]: IteratorRandom::choosel, ׮ , , - ׮$,4dk(*$1   =u7=: Uniformly sample `amount` distinct elements into a buffer77FC Collects values at random from the iterator into a supplied buffer8  until that buffer is filled.88IF Although the elements are selected randomly, the order of elements in9HE the buffer is neither stable nor fully random. If random ordering is9  desired, shuffle the result.::NK Returns the number of elements added to the buffer. This equals the length:NK of the buffer unless the iterator contains insufficient elements, in which;63 case this equals the number of elements available.; Complexity is `O(n)` where `n` is the length of the iterator.<:7 For slices, prefer [`IndexedRandom::choose_multiple`].= ׮  = = = ׮$=4``13HqB>; Uniformly sample `amount` distinct elements into a [`Vec`]CCLI This is equivalent to `choose_multiple_fill` except for the result type.CCI&DH&E &EEIF The length of the returned vector equals `amount` unless the iteratorFIF contains insufficient elements, in which case it equals the number ofF elements available.FGA&G:&H H H8`] may implement this (only [`Self::len`] must be| specified).l9d.':'*t The length  $  True when the length is zeroD  $ \ &  B? Returns a reference to one uniformly-sampled random element of /, the slice, or `None` if the slice is empty.  %" For slices, complexity is `O(1)`. l  <  ! use rand::seq::IndexedRandom;  '$ let choices = [1, 2, 4, 8, 16, 32];  /, println!("{:?}", choices.choose(&mut rng)); 41 assert_eq!(choices[..0].choose(&mut rng), None);< 4   ׮92g      ׮$ 4U(*96 Uniformly sample `amount` distinct elements from selfKH Chooses `amount` elements from the slice at random, without repetition,FC and in random order. The returned iterator is appropriate both forIF collection into a `Vec` and filling an existing buffer (see example).IF In case this API is not sufficiently flexible, use [`index::sample`].<9 For slices, complexity is the same as [`index::sample`].l<!'# let mut rng = &mut rand::rng();/, let sample = "Hello, audience!".as_bytes();)& // collect the results into a vector:LI let v: Vec = sample.choose_multiple(&mut rng, 3).cloned().collect();̃ // store in a buffer:ܡ let mut buf = [0u8; 5];VS for (b, slot) in sample.choose_multiple(&mut rng, buf.len()).zip(buf.iter_mut()) { *slot = *b;,<  8< 7<|  ׮Ȑ%6  ,  ׮$4(V(VBDFC Uniformly sample a fixed-size array of distinct elements from selfFC Chooses `N` elements from the slice at random, without repetition, and in random order.B? For slices, complexity is the same as [`index::sample_array`].l<!'#'/'EB let a: [u8; 3] = sample.choose_multiple_array(&mut rng).unwrap();<  ׮  tי',,  ׮%O4QTVS <    * # Biased sampling for one element!!FC Returns a reference to one element of the slice, sampled according!GD to the provided weights. Returns `None` only if the slice is empty.""DA The specified function `weight` maps each item `x` to a relative"JG likelihood `weight(x)`. The probability of each item being selected is#GD therefore `weight(x) / s`, where `s` is the sum of all `weight(x)`.$$30 For slices of length `n`, complexity is `O(n)`.$85 For more information about the underlying algorithm,%+( see the [`WeightedIndex`] distribution.%%%" See also [`choose_weighted_mut`].%l&&<&̧&&&;8 let choices = [('a', 2), ('b', 1), ('c', 1), ('d', 0)];''QN // 50% chance to print 'a', 25% chance to print 'b', 25% chance to print 'c',($! // and 'd' will never be printed(RO println!("{:?}", choices.choose_weighted(&mut rng, |item| item.1).unwrap().0);<))%" [`choose`]: IndexedRandom::choose)B? [`choose_weighted_mut`]: IndexedMutRandom::choose_weighted_mut*<9 [`WeightedIndex`]: crate::distr::weighted::WeightedIndex* * *8<* *7<**|*  ׮©ٳܳݳ޳߳i=2* G% + * * * * + © ,©ϭ'  ,|,l,4,l, ׮©L4P1P-5-l-d-,-   ;.1. Biased sampling of `amount` distinct elements//DA Similar to [`choose_multiple`], but where the likelihood of each/EB element's inclusion in the output may be specified. Zero-weighted0GD elements are never returned; the result may therefore contain fewer0IF elements than `amount` even when `self.len() >= amount`. The elements141 are returned in an arbitrary, unspecified order.11D'2J'3G΢'33MJ This implementation uses `O(length + amount)` space and `O(length)` time.4/, See [`index::sample_weighted`] for details.4l45<5̗5551. let choices = [('a', 2), ('b', 1), ('c', 1)];56,) // First Draw * Second Draw = total odds6 // -----------------------6^[ // (50% * 50%) + (25% * 67%) = 41.7% chance that the output is `['a', 'b']` in some order.7^[ // (50% * 50%) + (25% * 67%) = 41.7% chance that the output is `['a', 'c']` in some order.8^[ // (25% * 33%) + (25% * 33%) = 16.6% chance that the output is `['b', 'c']` in some order.9pm println!("{:?}", choices.choose_multiple_weighted(&mut rng, 2, |item| item.1).unwrap().collect::>());<::74 [`choose_multiple`]: IndexedRandom::choose_multiple; ; ;8<; ;7,;;ā<  ׮©ٳܳݳ޳߳i=2*Ȑ%6% < < < <י',= = © >©'  >L> ׮©)J4NN1N,@   C;AJ'ABGD This trait is implemented automatically for every type implementingB85 [`IndexedRandom`] and [`std::ops::IndexMut`].CClC |C='1'$O'C'6EhC&# Uniformly sample one element (mut)CCJG Returns a mutable reference to one uniformly-sampled random element ofD/ʼn'DE%'TE  ׮92g  E E F ׮$E4[K(*NG)& Biased sampling for one element (mut)GGNK Returns a mutable reference to one element of the slice, sampled accordingHGӠ'IID'IJ'JG΢'KK3'K8'L+'LL! See also [`choose_weighted`].LL0- [`choose_mut`]: IndexedMutRandom::choose_mutM74 [`choose_weighted`]: IndexedRandom::choose_weightedM<'N N N8; let mut bytes = "Hello, random!".to_string().into_bytes();U bytes.shuffle(&mut rng);U0- let str = String::from_utf8(bytes).unwrap();ĀV println!("{}", str); self.len()/2`_.+ then you may improve performance by taking_C@ `amount = self.len() - amount` and using only the second slice.``IF If `amount` is greater than the number of elements in the slice, thisa  will perform a full shuffle.aa85 For slices, complexity is `O(m)` where `m = amount`.|a  ׮  ' b bי',c c ׮$b4b4b9;c ccc  c $cdJJdld |d+-d d eHv   v-Lv  v "ݮ$vwnw w w8`. Conversion may or may not be trivial.D       &<9 Iterate over the indices as a sequence of `usize` values$ U% [  $JJJlDL$&MJ Convert into an iterator over the indices as a sequence of `usize` valuesL &    $ ,tD    ݋D   ݋   $  ! ! ! !8! !7ʓ#$! #!$! 4#"8KKKL$$$$#$$  $ 8$$$%   &   я&-L&  & 8$&(+8*"KKKL+$++($+ 92gK + $++,   -   --L-  - $-./+8b0KH Randomly sample exactly `amount` distinct indices from `0..length`, and01. return them in random order (fully shuffled).11LI This method is used internally by the slice sampling methods, but it can1MJ sometimes be useful to have the indices themselves so this is provided as2 an alternative.22IF The implementation used is not specified; we automatically select the3HE fastest available algorithm for the `length` and `amount` parameters3LI (based on detailed profiling on an Intel Haswell CPU). Roughly speaking,4NK complexity is `O(amount)`, except that when `amount` is small, performance4KH is closer to `O(amount^2)`, and when `length` is close to `amount` then5 `O(length)`.55NK Note that performance is significantly better over `u32` indices than over6HE `u64` indices. Because of this we hide the underlying type behind an6 abstraction, `IndexVec`.77HE If an allocation-free `no_std` function is required, it is suggested785 to adapt the internal `sample_floyd` implementation.88  Panics if `amount > length`.R|848  8 89 4WWW/1? ? ? ?B B BJE>; Randomly sample `amount` distinct indices from `0..length`EENK The result may contain less than `amount` indices if insufficient non-zeroFOL weights are available. Results are returned in an arbitrary order (there isF+( no guarantee of shuffling or ordering).GGGD Function `weight` is called once for each index to provide weights.GGLߌ(HM(H(II$IQN - [`WeightError::InvalidWeight`] when a weight is not-a-number or negative.IIM'J J J8 len`.d    t 4ul tl358 $   ġU-%" Access the thread-local generator'$ Use [`rand::rng()`](rng()) instead.  8< 7Rd%Renamed to `rng`9T%տ( VVVV-VVVV$CKH Generate a random value using the thread-local random number generator.PM This function is shorthand for [rng()].[random()](Rng::random):IF - See [`ThreadRng`] for documentation of the generator and securityVS - See [`StandardUniform`] for documentation of supported types and distributionst # Examples<! let x = rand::random::();$" let y = rand::random::(); println!("{}", y);.+ if rand::random() { // generates a boolean ,) println!("Better lucky than good!");, <   IF If you're calling `random()` repeatedly, consider using a local `rng`!96 handle to save an initialization-check on each usage:! let mut rng = rand::rng(); // a local handle to the generator"" let mut v = vec![1, 2, 3];#ܚ# for x in v.iter_mut() {Զ# *x = rng.random();,#<###/$$" [`ThreadRng`]: rngs::ThreadRng$ $ $8<$ $7Rd$$4$ $ |% 'w%1. Return an iterator over [`random()`] variates%%" This function is shorthand for&;8 [rng()].[random_iter](Rng::random_iter)().&l&&<&&<9 let v: Vec = rand::random_iter().take(5).collect();' println!("{v:?}");<'' ' '8<' '7Rd''\' ( |( .{)^[ Generate a random value in the given range using the thread-local random number generator.))"(*MJ [rng()].[random_range](Rng::random_range)(range).*l**<**/, let y: f32 = rand::random_range(0.0..=1e9);+(++IF let words: Vec<&str> = "Mary had a little lamb".split(' ').collect();,=: println!("{}", words[rand::random_range(..words.len())]);<,,KH Note that the first example can also be achieved (without `collect`'ing-63 to a `Vec`) using [`seq::IteratorRandom::choose`].- - -8<- -7Rd--d.׮ .ȇ .#.׮. ׮e@o=?BD2"/7%//"(/GD [rng()].[random_bool](Rng::random_bool)(p).0l00<001. println!("{}", rand::random_bool(1.0 / 3.0));<11d1%1Դ1%1 1 18<1 17Rd11R|1\2 o9=2H%L3֌%33H%3Jč%4F%4&%55C%5D%66"(6if [rng()].[random_ratio](Rng::random_ratio)(numerator, denominator).7d7%777%8l88<88-* println!("{}", rand::random_ratio(2, 3));<888#%9 9 98<9 97Rd99R|9d9 ןEqEq>+:8%::"(;<9 [rng()].[fill](Rng::fill)(dest).;l;;<;;%< rand::fill(&mut arr[..]);<<<$>  > >$> r&( fc The [Bernoulli distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution) `Bernoulli(p)`.OL This distribution describes a single boolean random variable, which is true<9 with probability `p` and false with probability `1 - p`.C@ It is a special case of the Binomial distribution with `n = 1`.T # PlotGD The following plot shows the Bernoulli distribution with `p = 0.1`, `p = 0.5`, and `p = 0.9`.mj ![Bernoulli distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/bernoulli.svg)l\ ```rust/, use rand::distr::{Bernoulli, Distribution};  )& let d = Bernoulli::new(0.3).unwrap(); '$ let v = d.sample(&mut rand::rng()); 74 println!("{} is from a Bernoulli distribution", v);<  |   FC This `Bernoulli` distribution uses 64 bits from the RNG (a `u64`), FC so only probabilities that are multiples of 2-64 can be represented.L `?T , rrrrr8rrrnrrnrr   ,  op`  ``` ]`]9ֹ`  , qq    L  0- Error type returned from [`Bernoulli::new`].tgi `p < 0` or `p > 1`.%EG"""""8"""r""r""   , s``t```  ``` ]`]9ֹ`  ,`uu   L ||v|||||8|||  ĺ!'$ An iterator over a [`Distribution`]HE This iterator yields random values of type `T` with distribution `D`(% from a random generator of type `R`.  B? Construct this `struct` using [`Distribution::sample_iter`] or EB [`Rng::sample_iter`]. It is also used by [`Rng::random_iter`] and܍! [`crate::random_iter`].$! !ȇ !ه !%ҏhAR1368;=D!,!"Ѯ4!!"׮!% This `struct` is created by the [`Distribution::map`] method.(# See its documentation for more.( (nj (ه ( (|~s9;>@CEHJD),)"Ѯ<)$)"Ư).<)"  x Anj @ه ? >) `Ư ) `Я `    ``` ]`]9ֹ`( ѮƯݮЯ,( NK A distribution to sample floating point numbers uniformly in the half-open2/ interval `(0, 1]`, i.e. including 1 but not 0.JG All values that can be generated are of the form `n * ε/2`. For `f32`MJ the 24 most significant random bits of a `u32` are used and for `f64` theIF 53 most significant bits of a `u64` are used. The conversion uses theԏ multiplicative method.IF See also: [`StandardUniform`] which samples from `[0, 1)`, [`Open01`]LI which samples from `(0, 1)` and [`Uniform`] which samples from arbitrary\  ranges. l <  ! " use rand::distr::OpenClosed01;  41 let val: f32 = rand::rng().sample(OpenClosed01); )& println!("f32 from (0, 1): {}", val);<   63 [`StandardUniform`]: crate::distr::StandardUniform $! [`Open01`]: crate::distr::Open01 /, [`Uniform`]: crate::distr::uniform::Uniformd  GJ2F58yy   ,   zMM{`MM  ``` ]`]9ֹ`  , M|  IF A distribution to sample floating point numbers uniformly in the open :7 interval `(0, 1)`, i.e. not including either endpoint.OL All values that can be generated are of the form `n * ε + ε/2`. For `f32`OL the 23 most significant random bits of an `u32` are used, for `f64` 52 from;8 an `u64`. The conversion uses a transmute-based method.OL See also: [`StandardUniform`] which samples from `[0, 1)`, [`OpenClosed01`]LI which samples from `(0, 1]` and [`Uniform`] which samples from arbitrary\)l<! use rand::distr::Open01;.+ let val: f32 = rand::rng().sample(Open01);))<6)0- [`OpenClosed01`]: crate::distr::OpenClosed01/Ӓ)4ðϛ%z5ð+++++8+++}ð++}ð++ ðð , ~ðiið`ii ð ``` ]`]9ֹ` ,iðð VW07 oy*(50    4y&($'(%50%  %%%  %4O&(+!ð(+50+ ð ,,,  ,4&( VW07 * (50    4&($' (%50%  %%%  %4u&(+! ð(+50+ ð ,,,  ,4^&(* (50    4&(* (50    4&(* (50    4J&(* (50    4&( *( 50        4&(!*(!50!  !!!  !4s&("*("50"  ߳"""  "43&(#*(#50#  ###  #4&($*($50$  ۴$$$  $4&(%*(%50%  %%%  %4j&(&*(&50&  ׵&&&  &41&('*('50'  '''  '4&((*((50(  Ӷ(((  (4&()*()50)  )))  )4&(**(*50*  Ϸ***  *4M&(HE Sample a `u8`, uniformly distributed over ASCII letters and numbers: a-z, A-Z and 0-9.l<!" use rand::distr::Alphanumeric; SP let chars: String = (0..7).map(|_| rng.sample(Alphanumeric) as char).collect(); (% println!("Random chars: {}", chars);<   FC The [`SampleString`] trait provides an easier method of generating >; a random [`String`], and offers more efficient allocation:<  2/ use rand::distr::{Alphanumeric, SampleString}; B? let string = Alphanumeric.sample_string(&mut rand::rng(), 16); *' println!("Random string: {}", string);<  |  # Passwords  OL Users sometimes ask whether it is safe to use a string of random characters NK as a password. In principle, all RNGs in Rand implementing `CryptoRng` areLI suitable as a source of randomness for generating passwords (if they areHE properly seeded), but it is more conservative to only use randomnessHE directly from the operating system via the `getrandom` crate, or the/, corresponding bindings of a crypto library.MJ When generating passwords or keys, it is important to consider the threatLI model and in some cases the memorability of the password. This is out ofFC scope of the Rand project, and therefore we defer to the following| references:_\ - [Wikipedia article on Password Strength](https://en.wikipedia.org/wiki/Password_strength)[X - [Diceware for generating memorable passwords](https://en.wikipedia.org/wiki/Diceware)d9`LcJ+5`  ``` ]`]9ֹ`+ ,ccccc8ccccccc , , - 85 Sample a [`u8`], uniformly distributed over letters: a-z and A-Z.lOL You're able to generate random Alphabetic characters via mapping or via the30 [`SampleString::sample_string`] method like so:<!0- use rand::distr::{Alphabetic, SampleString}; // Manual mappingQN let chars: String = (0..7).map(|_| rng.sample(Alphabetic) as char).collect();(),) // Using [`SampleString::sample_string`]@= let string = Alphabetic.sample_string(&mut rand::rng(), 16);*)<|)(% Refer to [`Alphanumeric#Passwords`].T?}Q&35`  ``` ]`]9ֹ`/ ,8 0 , 1 9GQ(9IA09I  9I9I9I  9I4&(;G\ P&;H(&(;IA0;I  ׮;I ;I ;I ׮ ;I4*,=G\G)( P)=H&(dg;==IA0=I  =I=I=I  =I4:.0?G\Gk))Wɺ P)?H) &ɺ(PR?IA0?I  ?I ?I ?I ɺ ?I424AG\Gk))) P)AH) )& (nqEHegAIA0AI  غ  AIŽ AIӎ AI ɺ AI468CG\ G k  ^      )))G) P)CH) ))&& ( ;>25z|CIA0CI     CI CICI ɺ CI4BDEG\ G k  ^        )))H)) P)EH) )))&, ( IL #@CEIA0EI     EI EIEI ɺ EI4GIGG\ G k  ^  _       ) ))z))*))ƻ P)GH) ))))!&ƻ2 (   _ b  GIA0GI     GI GIGI ɺƻ GI4KMIG\ G k  ^  _ _        )m )< ) )L))*Y P)IH) ))))!Ձ*'&8 (     v y 69=@IIA0II  ֻ   II IIII ɺƻ II4OQKG \ G k  ^  _ _ 0         )) )n ) ) )** P)KH) ))))!Ձ*'Ć*-&> ( mp      X[KIA0KI     KI KIKI ɺƻ KI4! SUMG \ G k  ^  _ _ 0 a         )))0) ) ) **K*ļ P)MH) ))))!Ձ*'Ć*-*3&ļD ( eh<?\_    '*MIA0MI     MI MIMI ɺƻļ MI4 WYOG \ G k  ^  _ _ 0 a b         )))"))) * *=** P)OH) ))))!Ձ*'Ć*-*3*9&J ( dg;>[^    & ) OIA0OI  ϞԼ   OI ϞOIϞOI ɺƻļϞ OI4[]QG \ G k  ^  _ _ 0 a b c         )))D)))* *_ ** * P)QH) ))))!Ձ*'Ć*-*3*9*?&P ( jmU X   '* QIA0QI  ͦ   QI ͦQIͦQI ɺƻļͦ QI4_aR  XR0R  R% R  S  XS0S  S% S  ;ET  XT0T  T% T  U   XU0U  U% U  !!V!  XV0V  V% V  ''U*PVW""  XW"0W"  W#' W#  ++XD#  HXD XD  HXD XD  XE$0XE  - -XF$0XF  5. ?.XGPXG  */XH+8XH  / /XI2xXI*  E1N1XJ'hXJ  `XO#_D$  H_D _D  H_D _D  _E$0_E  2 2_F$0_F  43 >3_GP_G  )4_H+8_H  4 4_I2x_I*  36<6_J'h_J  y_O$Ć:7 A distribution to uniformly sample elements of a sliceNK Like [`IndexedRandom::choose`], this uniformly samples elements of a sliceNK without modification of the slice (so called "sampling with replacement").NK This distribution object may be a little faster for repeated sampling (but)& slower for small numbers of samples).| ## Examples:7 Since this is a distribution, [`Rng::sample_iter`] and;8 [`Distribution::sample_iter`] may be used, for example:<30 use rand::distr::{Distribution, slice::Choose};+( let vowels = ['a', 'e', 'i', 'o', 'u']; 41 let vowels_dist = Choose::new(&vowels).unwrap();  " // build a string of 10 vowels *' let vowel_string: String = vowels_dist &# .sample_iter(&mut rand::rng())  .take(10)   ! println!("{}", vowel_string); '$ assert_eq!(vowel_string.len(), 10); ?< assert!(vowel_string.chars().all(|c| vowels.contains(&c)));<   DA For a single sample, [`IndexedRandom::choose`] may be preferred:<  !'  +*   52 println!("{}", vowels.choose(&mut rng).unwrap());<   @= [`IndexedRandom::choose`]: crate::seq::IndexedRandom::choose1. [`Rng::sample_iter`]: crate::Rng::sample_iter%4"    "tpz 684,  !,  \  77"    `7 ` "77  ``` ]`]9ֹ` f ",799"   99  +.ad99  g ", h"    O7: Error: empty sliceHE This error is returned when [`Choose::new`] is given an empty slice.',hFNz/k 4  JJ ` JJ  ``` ]`]9ֹ` o ,JJJJJJ8JJJJJ  JJ  p , qԤJG The back-end implementing [`UniformSampler`] for floating-point types.OL Unless you are implementing [`UniformSampler`] for your own type, this type96 should not be used directly, use [`Uniform`] instead.Գ # Implementation notesMJ `UniformFloat` implementations convert RNG output to a float in the range NK `[1, 2)` via transmutation, map this to `[0, 1)`, then scale and translate NK to the desired range. Values produced this way have what equals 23 bits of 30 random digits for an `f32` and 52 for an `f64`.   # Bias and range errors  MJ Bias may be expected within the least-significant bit of the significand. MJ It is not guaranteed that exclusive limits of a range are respected; i.e. KH when sampling the range `[a, b)` it is not guaranteed that `b` is neverd  sampled.   [`new`]: UniformSampler::new 41 [`new_inclusive`]: UniformSampler::new_inclusive 6) [`Uniform`]: super::Uniform)d%  # 0w  !4  D,  __%  #פ$_  __  t , u%  #nB``%  #`` ` ``  ``` ]`]9ֹ` v ,`bb%  #bb%  #8b  ^abb   w L bx* x;Xx  gx xg(xx* `x8x  x)*  0xx  xx % %   #(xxŘ#(x*x g gxxx!hx! % %   #(x"*x"Ř#(x#*x" kkx!x!x(90x(  *   x) x(x(  x)4Zo')x/hx/ ٳܳݳ޳߳i=2**  % %   x0 x/x/#(x0*x0#(x1*x1 oqzq4quwx/x/x2x2 *  % %   x3 x2x2#(x3*x3#(x4*x4 8vCv4Nv]_x2x2}+ };X}  g} }g(}}+ `}8}  })+  0}}  }} % %    #(}}Ř#(}*} |%|}}}!h}! % %    #(}"*}"Ř#(}#*}" ݀}!}!}(90}(  *   }) }(}(  })4r')}/h}/ ٳܳݳ޳߳i=2**  % %    }0 }/}/#(}0*}0#(}1*}1 4uw}/}/}2}2 *  % %   }3 }2}2#(}3*}3#(}4*}4 Ze4p]_}2}2C@ The back-end implementing [`UniformSampler`] for integer types.O*9*԰*JG For simplicity, we use the same generic struct `UniformInt` for all PM integer types `X`. This gives us only one field type, `X`; to store unsigned PM values of this size, we take use the fact that these conversions are no-ops.  LI For a closed range, the number of possible numbers we should generate is NK `range = (high - low + 1)`. To avoid bias, we must ensure that the size of LI our sample space, `zone`, is a multiple of `range`; other values must be 52 rejected (by replacing with a new random sample).  LI As a special case, we use `range = 0` to represent the full range of the ?< result type (i.e. for `new_inclusive($ty::MIN, $ty::MAX)`).  JG The optimum `zone` is the largest product of `range` which fits in ourPM (unsigned) target type. We calculate this by calculating how many numbers weVS must reject: `reject = (MAX + 1) % range = (MAX - range + 1) % range`. Any (large)MJ product of `range` will suffice, thus in `sample_single` we multiply by aGD power of 2 via bit-shifting (faster but may cause more rejections).NK The smallest integer PRNGs generate is `u32`. For 8- and 16-bit outputs weMJ use `u32` for our `zone` and samples (because it's not slower and becauseOL it reduces the chance of having to reject a sample). In this case we cannotJG store `zone` in the target type since it is too large, however we know*' `ints_to_reject < range <= $uty::MAX`.LI An alternative to using a modulus is widening multiply: After a wideningOL multiply by `range`, the result is in the high word. Then comparing the low?< word against `zone` makes sure our distribution is uniform.T # BiasNK Unless the `unbiased` feature flag is used, outputs may have a small bias.A"MJ 56 (`i8` and `u8`), 48 (`i16` and `u16`), 96 (`i32` and `u32`), 64 (`i64`(% and `u64`), 128 (`i128` and `u128`).*,T%  # ?R%_Ps !#  ,  L4  ii%  #פ$i  ii   , %  #*;ϏϏ%  #*Ϗ ` ϏϏ  ``` ]`]9ֹ`  ,Ϗ%  # %  #U*  or    L %  #0  RU8   Ї- `8  '-  0  ʭ % %   #( Ř#( +  #h#ʭ % %    #($+$Ř#(%+% +##+9+7"0+  +   + ++  ,4*,0h0 ٳܳݳ޳߳i=2*+  % %   1 00#(2+2#(2+2 ԥ4{}0075/"55A"6@"6678 66788667"P66!7 ݜ+  % %    8 77#(9+9#(9+9 4ce77Ј. `8  '.  0   % %    #( Ř#( å+  #h# % %   #($+$Ř#(%ߥ+% ##+9+7"0+  å+   + ++  ,4n*,0h0 ٳܳݳ޳߳i=2*å+  % %   1 00#(2å+2#(2å+2 4{}0075/"55A"6@"6678 66788667"P66!7 +  % %   8 77#(9ϫ+9#(9+9 8D4Pce77Љ/ `8  '/  0   % %   #( Ř#( +  #h# % %   #($+$Ř#(%+% ##+9+7"0+  +   + ++  ,4*,0h0 ٳܳݳ޳߳i=2*+  % %    1 00#(2+2#(2+2 4'{}0075/"55A"6@"6678 66788667"P66!7 ѷ+  % %   8 77#(9+9#(9+9 4ce77Њ0 `8  '0  0   % %    #( Ř#( +  #h# % %   #($+$Ř#(%+% 4@##+9+7"0+  +   + ++  ,4*,0h0 ٳܳݳ޳߳i=2*+  % %   1 00#(2+2#(2+2 4{}0075/"55A"6@"6678 66788667"P66!7 +  % %    8 77#(9+9#(9+9   4 ce77Ћ1 `8  '1  0  ͦ % %   #( Ř#( +  /;#h#ͦ % %    #($+$Ř#(%+% ##+9+7"0+  +   + ++  ,4*,0h0 ٳܳݳ޳߳i=2*+  % %   1 00#(2+2#(2+2 !!4!{}0075/"55A"6@"6678 66788667"P66!7 +  % %   8 77#(9+9#(9+9 **4*ce77Ќ2 `8  '2  0   % %   #( Ř#( +  00#h# % %   #($+$Ř#(%+% 55##+9+7"0+  +   + ++  ,4:*,0h0 ٳܳݳ޳߳i=2*+  % %    1 00#(2+2#(2+2 >>4>{}0075/"55A"6@"6678 66788667"P66!7 +  % %   8 77#(9+9#(9+9 bHnH4zHce77Ѝ3 `8  '3  0  ߝ  % %    #( Ř#( +  NN#h#ߝ  % %    #($+$Ř#(%+% SS##+9+7"0+  +   + ++  ,4X*,0h0 ٳܳݳ޳߳i=2*+  % %   1 00#(2+2#(2+2 ]$]40]{}0075/"55A"6@"6678 66788667"P66!7 +  % %    8 77#(9+9#(9+9 ff4fce77Ў4 `8  '4  0   % %   #( Ř#( +  ll#h# % %    #($+$Ř#(%+% gqsq##+9+7"0+  +   + ++  ,4v*,0h0 ٳܳݳ޳߳i=2*+  % %   1 00#(2+2#(2+2 4z@z4Lz{}0075/"55A"6@"6678 66788667"P66!7 +  % %   8 77#(9+9#(9+9 y4ce77Џ5 `8  '5  0    % %   #( Ř#( ك,  ׉#h#  % %   #($΃,$Ř#(%,% ##+9+7"0+  ك,   + ++  ,4[*,0h0 ٳܳݳ޳߳i=2*ك,  % %   1 00#(2ك,2#(2ك,2 |4{}0075/"55A"6@"6678 66788667"P66!7 ,  % %   8 77#(9,9#(9,9 ͠٠4ce77А6 `8  '6  0    % %   #( Ř#( ,  CO#h#  % %   #($,$Ř#(%,% .:##+9+7"0+  ,   + ++  ,4s*,0h0 ٳܳݳ޳߳i=2*,  % %    1 00#(2,2#(2,2 S_4k{}0075/"55A"6@"6678 66788667"P66!7 ,  % %   8 77#(9,9#(9,9 ̼ؼ4ce77=: The back-end implementing [`UniformSampler`] for `usize`.*LI Sampling a `usize` value is usually used in relation to the length of anMJ array or other memory structure, thus it is reasonable to assume that theKH vast majority of use-cases will have a maximum size under [`u32::MAX`].LI In part to optimise for this use-case, but mostly to ensure that resultsςPM are portable across 32-bit and 64-bit architectures (as far as is possible),?< this implementation will use 32-bit sampling when possible.  8  8 7$$ 8 7ʓ#$E7d/I T  dɉ,ɉ  l4  d  8 7ʓ#$#4  99999899999  99   ,   `   ``` ]`]9ֹ`  ,     Ä LÄ oo  ooooo8ooo  ΄ ΄ < 9 The back-end implementing [`UniformSampler`] for `char`. O *9 * N K This differs from integer range sampling since the range `0xD800..=0xDFFF`L I are used for surrogate pairs in UCS and UTF-16, and consequently are notN K valid Unicode code points. We must therefore avoid sampling values in thisT  range.;\ ( ľ <   8      ,   `   ``` ]`]9ֹ`   ,              L     8     ! @ = The back-end implementing [`UniformSampler`] for `Duration`. P M Unless you are implementing [`UniformSampler`] for your own types, this type 9 *<|! %&Q{! ̅" $"   \" 4"   jjjjj8jjjjj  jj  !  ,! ! @ @  ` @ @   ``` ]`]9ֹ` !  ,! @ z z z z   z z    !  L!  z          8     !  ! Ġ# =# FJ    ,# =  L# $#   # ,#   4$ =  $ ,$   Κ ,$ =  l$ D$   t$ L$   $ $$   Κ O O  ` O O   ``` ]`]9ֹ` "  ," O  " <<<<<8<<<<<  <<  "  ,"      "  L"    8  "  " t" !B ? Error type returned from [`Uniform::new`] and `new_inclusive`.>," }}j   T" "6 3 `low > high`, or equal in case of exclusive range.>E  eg  L# #M J Input or range `high - low` is non-finite. Not relevant to integer types.>\  |~  [[[[[8[[[[[  [[  "  ," " [[ ` [[  ``` ]`]9ֹ` "  ," [\\\\  \\   "  L"  \]]  ]]]]]8]]]  "  " i'/ , Sample values uniformly between two bounds.' '  # Construction' 'G D [`Uniform::new`] and [`Uniform::new_inclusive`] construct a uniform(O L distribution sampling from the given `low` and `high` limits. `Uniform` may)R O also be constructed via [`TryFrom`] as in `Uniform::try_from(1..=6).unwrap()`.) )P M Constructors may do extra work up front to allow faster sampling of multiple*I F values. Where only a single sample is required it is suggested to use*H E [`Rng::random_range`] or one of the `sample_single` methods instead.+ +H E When sampling from a constant range, many calculations can happen at,N K compile-time and all methods should be fast; for floating-point ranges and,O L the full range of integer types, this should have comparable performance to-A > the [`StandardUniform`](super::StandardUniform) distribution.- -  # Provided implementations. .Q N - `char` ([`UniformChar`]): samples a range over the implementation for `u32`.O L - `f32`, `f64` ([`UniformFloat`]): samples approximately uniformly within a/P M range; bias may be present in the least-significant bit of the significand0H E and the limits of the input range may be sampled even when an open0  (exclusive) range is used0J G - Integer types ([`UniformInt`]) may show a small bias relative to the1N K expected uniform distribution of output. In the worst case, bias affects2P M 1 in `2^n` samples where n is 56 (`i8` and `u8`), 48 (`i16` and `u16`), 962G D (`i32` and `u32`), 64 (`i64` and `u64`), 128 (`i128` and `u128`).32 / The `unbiased` feature flag fixes this bias.3F C - `usize` ([`UniformUsize`]) is handled specially, using the `u32`4P M implementation where possible to enable portable results across 32-bit and4  64-bit CPU architectures.5O L - `Duration` ([`UniformDuration`]): samples a range over the implementation5  for `u32` or `u64`5K H - SIMD types (requires [`simd_support`] feature) like x86's [`__m128i`]6G D and `std::simd`'s [`u32x4`], [`f32x4`] and [`mask32x4`] types are7I F effectively arrays of integer or floating-point types. Each lane is7O L sampled independently from its own range, potentially with more efficient8G D random-bit-usage than would be achieved with sequential sampling.8 l8 9 <9 9- 9 98 5 let between = Uniform::try_from(10..10000).unwrap();9 :  let mut sum = 0;:  for _ in 0..1000 {:( % sum += between.sample(&mut rng);,: :  println!("{}", sum);<; ; ;@ = For a single sample, [`Rng::random_range`] may be preferred:; <; ; !; ; <, ) println!("{}", rng.random_range(0..10));<< < <  [`new`]: Uniform::new<- * [`new_inclusive`]: Uniform::new_inclusive=, ) [`Rng::random_range`]: Rng::random_range=L I [`__m128i`]: https://doc.rust-lang.org/core/arch/x86/struct.__m128i.html>  [`u32x4`]: std::simd::u32x4>  [`f32x4`]: std::simd::f32x4>% " [`mask32x4`]: std::simd::mask32x4>H E [`simd_support`]: https://github.com/rust-random/rand#crate-features?`] exists. The implementation guarantees thatI F elements with zero weight are never picked, even when the weights are  floating point numbers.   # Performance  H E Time complexity of sampling from `WeightedIndex` is `O(log N)` where !  `N` is the number of weights. P $ K $  L I A `WeightedIndex` contains a `Vec` and a [`Uniform`] and so its O L size is the sum of the size of those objects, possibly plus some alignment.  L I Creating a `WeightedIndex` will allocate enough space to hold `N - 1` K H weights of type `X`, where `N` is the number of weights. However, sinceK H `Vec` doesn't guarantee a particular growth strategy, additional memoryJ G might be allocated but not used. Since the `WeightedIndex` object alsoR O contains an instance of `X::Sampler`, this might cause additional allocations,K H though for primitive types, [`Uniform`] doesn't allocate any memory. A > Sampling from `WeightedIndex` will result in a single call toP M `Uniform::sample` (method of the [`Distribution`] trait), which typicallyK H will request a single value from the underlying [`RngCore`], though theG D exact number depends on the implementation of `Uniform::sample`. l  <  - ǰ$ "  let choices = ['a', 'b', 'c'];  let weights = [2, 1, 1];5 $   for _ in 0..100 {T Q // 50% chance to print 'a', 25% chance to print 'b', 25% chance to print 'c'7 4 println!("{}", choices[dist.sample(&mut rng)]);,  5 2 let items = [('a', 0.0), ('b', 3.0), ('c', 7.0)];M J let dist2 = WeightedIndex::new(items.iter().map(|item| item.1)).unwrap(); -S P // 0% chance to print 'a', 30% chance to print 'b', 70% chance to print 'c'8 5 println!("{}", items[dist2.sample(&mut rng)].0);, <  ) & [`Uniform`]: crate::distr::Uniform %^ ܎$Fl %  #  һ#l $T  13    | d       %  #*һ#$, `   ``` ]`]9ֹ`   , %  #פ$һ#&$%,  OR    , bb%  #һ#$ bb%  #P*bһ#W$V,b  bb     L  b         <                 <        f"         <        2&         <        *         <        -         <        2         <        5         <        8         <         <         <        >         <        A         <        DI   <     FJ <    Ft̷ Invalid weight errors<9 This type represents errors from [`WeightedIndex::new`],GD [`WeightedIndex::update_weights`] and other weighted distributions.2K,R=3dDA The input weight sequence is empty, too long, or wrongly orderedKSsulOL A weight is negative, too large for the distribution, or not a valid numberK^~>; Not enough non-zero weights are available to sample valuesJG When attempting to sample a single value this implies that all weightsIF are zero. When attempting to sample `amount` values this implies that52 less than `amount` weights are greater than zero.K;\_D0- Overflow when calculating the sum of weightsK?_aGG`GG  ``` ]`]9ֹ` ,GsHsHsHsHsH8sHsHsHsHsHsHsH  , HHHHHH   L HIIIIIII8III  ԲG$%" The Standard Uniform distribution%%SP This [`Distribution`] is the *standard* parameterization of [`Uniform`]. Bounds%.+ are selected according to the output type.&&FC Assuming the provided `Rng` is well-behaved, these implementations&@= generate values with the following ranges and distributions:''C@ * Integers (`i8`, `i32`, `u64`, etc.) are uniformly distributed'OL over the whole range of the type (thus each possible value may be sampled( with equal probability).(NK * `char` is uniformly distributed over all Unicode scalar values, i.e. all)DA code points in the range `0...0x10_FFFF`, except for the range)B? `0xD800...0xDFFF` (the surrogate code points). This includes*&# unassigned/reserved code points.*UR For some uses, the [`Alphanumeric`] or [`Alphabetic`] distribution will be more+ appropriate.+B? * `bool` samples `false` or `true`, each with probability 0.5.+MJ * Floating point types (`f32` and `f64`) are uniformly distributed in the,[X half-open range `[0, 1)`. See also the [notes below](#floating-point-implementation).-NK * Wrapping integers ([`Wrapping`]), besides the type identical to their- normal integer variants..LI * Non-zero integers ([`NonZeroU8`]), which are like their normal integer.&# variants but cannot sample zero.//PM The `StandardUniform` distribution also supports generation of the following/;8 compound types where all component types are supported:00JG * Tuples (up to 12 elements): each element is sampled sequentially and0OL independently (thus, assuming a well-behaved RNG, there is no correlationĹ1 between elements).1EB * Arrays `[T; n]` where `T` is supported. Each element is sampled2JG sequentially and independently. Note that for small `T` this usually2MJ results in the RNG discarding random bits; see also [`Rng::fill`] which3KH offers a more efficient approach to filling an array of integer types3 with random data.4KH * SIMD types (requires [`simd_support`] feature) like x86's [`__m128i`]4G,5I,5MJ sampled independently, potentially with more efficient random-bit-usage6NK (and a different resulting value) than would be achieved with sequential7/, sampling (as with the array types above).77 ## Custom implementations77VS The [`StandardUniform`] distribution may be implemented for user types as follows:8<88 # #![allow(dead_code)]8!852 use rand::distr::{Distribution, StandardUniform};99 struct MyF32 {|9 x: f32,,9992/ impl Distribution for StandardUniform {:A> fn sample(&self, rng: &mut R) -> MyF32 {:%" MyF32 { x: rng.random() }L:!,;<;;; ## Example usage<;̲;;%" use rand::distr::StandardUniform;;;74 let val: f32 = rand::rng().sample(StandardUniform);<)& println!("f32 from [0, 1): {}", val);<<<<# # Floating point implementation=WT The floating point implementations for `StandardUniform` generate a random value in=@= the half-open interval `[0, 1)`, i.e. including 0 but not 1.>>J)>M)?IÍ)@FC multiplicative method: `(rng.gen::<$uty>() >> N) as $ty * (ε/2)`.@@LI See also: [`Open01`] which samples from `(0, 1)`, [`OpenClosed01`] whichAOL samples from `(0, 1]` and `Rng::random_range(0..1)` which also samples fromAOL `[0, 1)`. Note that `Open01` uses transmute-based methods which yield 1 bitBPM less precision but may perform faster on some architectures (on modern IntelC;8 CPUs all methods have approximately equal performance).CC! [`Uniform`]: uniform::UniformC'$ [`Wrapping`]: std::num::WrappingD&# [`NonZeroU8`]: std::num::NonZeroU8DL,E,E,E%,EH,L|GLxR7s5IIIII8IIIIIII F ,F F"J"J`"J"J  ``` ]`]9ֹ`F ,F"JJJJJ \  \0 \    \ \ \  \ 4Wq*,\  \0 \    \ \ \  \ 4<{*,\ \0 \   \ \ \  \ 4*,\ \0 \   \ \ \  \ 4̎*,b b0 b   c b c  c 4*,k k0 k   k k k  k 4P*,b b0 b   c b c  c 4*,k k0 k   k k k  k 4`*,b b0 b   c b c  c 4h*,k k0 k   k k k  k 4*,b b0 b   c b c  c 4*,k k0 k   k k k  k 4>*,b b0 b   c b c  c 4 *,k k0 k   k k k  k 4*,b b0 b   c b c  c 4$*,k k0 k   k k k  k 482*,b b0 b   c b c  c 4[?*,k k0 k   k k k  k 4L*,b b0 b   c b c  c 4Y*,k k0 k   k k k  k 4dg*,b b0 b   c b c  c 4t*,k k0 k   k k k  k 4*,M J A wrapper around any PRNG that implements [`BlockRngCore`], that adds the  ability to reseed it. F C `ReseedingRng` reseeds the underlying PRNG in the following cases: ' $ - On a manual call to [`reseed()`].? < - After `clone()`, the clone will be reseeded on first use.I F - After the PRNG has generated a configurable number of random bytes. L I # When should reseeding after a fixed number of generated bytes be used? G D Reseeding after a fixed number of generated bytes is never strictly N K *necessary*. Cryptographic PRNGs don't have a limited number of bytes they P M can output, or at least not a limit reachable in any practical way. There is . + no such thing as 'running out of entropy'.  P M Occasionally reseeding can be seen as some form of 'security in depth'. Even P M if in the future a cryptographic weakness is found in the CSPRNG being used, G D or a flaw in the implementation, occasionally reseeding should make 9 6 exploiting it much more difficult or even impossible.  L I Use [`ReseedingRng::new`] with a `threshold` of `0` to disable reseeding , ) after a fixed number of generated bytes.   # Error handling O L Although unlikely, reseeding the wrapped PRNG can fail. `ReseedingRng` willF C never panic but try to handle the error intelligently through some? < combination of retrying and delaying reseeding until later.N K If handling the source error fails `ReseedingRng` will continue generating1 . data from the wrapped PRNG without reseeding. N K Manually calling [`reseed()`] will not have this retry or delay logic, but  reports the error. l  < ̡ E B use rand_chacha::ChaCha20Core; // Internal part of ChaChaRng that; 8 // implements BlockRngCoreԽ  use rand::rngs::OsRng;!  use rand::rngs::ReseedingRng; T Q let mut reseeding_rng = ReseedingRng::::new(0, OsRng).unwrap(); 2 / println!("{}", reseeding_rng.random::()); / , let mut cloned_rng = reseeding_rng.clone();I F assert!(reseeding_rng.random::() != cloned_rng.random::());<  4 1 [`BlockRngCore`]: rand_core::block::BlockRngCore, ) [`ReseedingRng::new`]: ReseedingRng::new& # [`reseed()`]: ReseedingRng::reseedSd 9%  %$ %d %\ %T ]uBA ȒȒ  Ȓ::9% %`:`:%>%=%<`nqux::  ``` ]`]9ֹ`  , :& Tl' 9% ' %$' e!++A4+-02D' ,' t' D' t' L' ' ' ::9%&%%.:.:`ORVY::  ``` ]`]9ֹ`&  ,& : 52 A mock generator yielding very predictable outputJG This generates an arithmetic sequence (i.e. adds a constant each step)IF over a `u64` number, using wrapping arithmetic. If the increment is 0$! the generator yields a constant.NK Other integer types (64-bit and smaller) are produced via cast from `u64`.OL Other types are produced via their implementation of [`Rng`](crate::Rng) or1. [`Distribution`](crate::distr::Distribution).LI Output values may not be intuitive and may change in future releases but are consideredDA [portable](https://rust-random.github.io/book/portability.html).96 (`bool` output is true when bit `1u64 << 31` is set.) l  <   # #![allow(deprecated)] ! " use rand::rngs::mock::StepRng;  (% let mut my_rng = StepRng::new(2, 1); +( let sample: [u64; 3] = my_rng.random(); " assert_eq!(sample, [2, 3, 4]);< &Deprecated without replacement GW< &.vLE4  &.4  &.&`&  ``` ]`]9ֹ`  , &&   , &&&    L  &8&    +63 A small-state, fast, non-crypto, non-portable PRNGPM This is the "standard small" RNG, a generator with the following properties:JG - Non-[portable]: any future library version may replace the algorithm,) and results may be platform-dependent.SP (For a small portable generator, use the [rand_pcg] or [rand_xoshiro] crate.)=: - Non-cryptographic: output is easy to predict (insecure)+( - [Quality]: statistically good qualityLI - Fast: the RNG is fast for both bulk generation and single values, with %" consistent cost of method calls̺  - Fast initialization IF - Small state: little memory usage (current state size is 16-32 bytes  depending on platform)   The current algorithm is OL `Xoshiro256PlusPlus` on 64-bit platforms and `Xoshiro128PlusPlus` on 32-bit EB platforms. Both are also implemented by the [rand_xoshiro] crate.   ## Seeding (construction)  HE This generator implements the [`SeedableRng`] trait. All methods are NK suitable for seeding, but note that, even with a fixed seed, output is not ! [portable]. Some suggestions:  PM 1. To automatically seed with a unique seed, use [`SeedableRng::from_rng`]:\ ``` use rand::SeedableRng;! use rand::rngs::SmallRng;74 let rng = SmallRng::from_rng(&mut rand::rng());  # let _: SmallRng = rng;\.(% or [`SeedableRng::from_os_rng`]:\.  # use rand::SeedableRng;# # use rand::rngs::SmallRng;*' let rng = SmallRng::from_os_rng(); .\.NK 2. To use a deterministic integral seed, use `seed_from_u64`. This uses aJG hash function internally to yield a (typically) good seed from anyt input.\.2/ # use rand::{SeedableRng, rngs::SmallRng};-* let rng = SmallRng::seed_from_u64(1); .\.PM 3. To seed deterministically from text or other input, use [`rand_seeder`].(% See also [Seeding RNGs] in the book. ## GenerationLI The generators implements [`RngCore`] and thus also [`Rng`][crate::Rng].5%D%IF [Seeding RNGs]: https://rust-random.github.io/book/guide-seeding.htmlI%IF [Quality]: https://rust-random.github.io/book/guide-rngs.html#quality# [`StdRng`]: crate::rngs::StdRng1. [rand_pcg]: https://crates.io/crates/rand_pcg96 [rand_xoshiro]: https://crates.io/crates/rand_xoshirofc [`rand_chacha::ChaCha8Rng`]: https://docs.rs/rand_chacha/latest/rand_chacha/struct.ChaCha8Rng.htmlDA [`rand_seeder`]: https://docs.rs/rand_seeder/latest/rand_seeder/XDs&z;C   ,  `    ``` ]`]9ֹ` ,    L ׸׸׸׸׸׸׸8׸׸׸  Y\  PP`PP  ``` ]`]9ֹ` ,PNNNN  ,   L 8   0- A strong, fast (amortized), non-portable RNGJG This is the "standard" RNG, a generator with the following properties:J.,.EB (For a portable version, use the [rand_chacha] crate directly.)LI - [CSPRNG]: statistically good quality of randomness and [unpredictable]KH - Fast ([amortized](https://en.wikipedia.org/wiki/Amortized_analysis)):NK the RNG is fast for bulk generation, but the cost of method calls is not2/ consistent due to usage of an output buffer.  PM The current algorithm used is the ChaCha block cipher with 12 rounds. Please MJ see this relevant [rand issue] for the discussion. This may change as new B? evidence of cipher security and performance becomes available.  .  PM This generator implements the [`SeedableRng`] trait. Any method may be used, MJ but note that `seed_from_u64` is not suitable for usage where security is PM important. Also note that, even with a fixed seed, output is not [portable].  HE Using a fresh seed **direct from the OS** is the most secure option:<  ,) # use rand::{SeedableRng, rngs::StdRng};$! let rng = StdRng::from_os_rng();Ԭ # let _: StdRng = rng;<<9 Seeding via [`rand::rng()`](crate::rng()) may be faster:<,ˍ/1. let rng = StdRng::from_rng(&mut rand::rng());/<PM Any [`SeedableRng`] method may be used, but note that `seed_from_u64` is notMJ suitable where security is required. See also [Seeding RNGs] in the book..L.5%D%I.PM [unpredictable]: https://rust-random.github.io/book/guide-rngs.html#securityI%wt [CSPRNG]: https://rust-random.github.io/book/guide-gen.html#cryptographically-secure-pseudo-random-number-generator74 [rand_chacha]: https://crates.io/crates/rand_chacha@= [rand issue]: https://github.com/rust-random/rand/issues/932[4& C   ,4:`dh  ``` ]`]9ֹ` ,$(   L &,V_chl8x~   -* A reference to the thread-local generator  LI This type is a reference to a lazily-initialized thread-local generator.HE An instance can be obtained via [`rand::rng()`][crate::rng()] or via [`ThreadRng::default()`].JG The handle cannot be passed between threads (is not `Send` or `Sync`).t&IF Security must be considered relative to a threat model and validationJG requirements. The Rand project can provide no guarantee of fitness for@= purpose. The design criteria for `ThreadRng` are as follows:FC - Automatic seeding via [`OsRng`] and periodically thereafter (seeIF ([`ReseedingRng`] documentation). Limitation: there is no automatic52 reseeding on process fork (see [below](#fork)).QN - A rigorusly analyzed, unpredictable (cryptographic) pseudo-random generator`] (see [the book on security](https://rust-random.github.io/book/guide-rngs.html#security)).=: The currently selected algorithm is ChaCha (12-rounds).(% See also [`StdRng`] documentation.C@ - Not to leak internal state through [`Debug`] or serialization implementations.IF - No further protections exist to in-memory state. In particular, theNK implementation is not required to zero memory on exit (of the process or/, thread). (This may change in the future.)GD - Be fast enough for general-purpose usage. Note in particular thatIF `ThreadRng` is designed to be a "fast, reasonably secure generator"=: (where "reasonably secure" implies the above criteria).KH We leave it to the user to determine whether this generator meets their=: security requirements. For an alternative, see [`OsRng`].T # ForkKH `ThreadRng` is not automatically reseeded on fork. It is recommended toPM explicitly call [`ThreadRng::reseed`] immediately after a fork, for example:l ```ignore fn do_fork() {*' let pid = unsafe { libc::fork() }; if pid == 0 {30 // Reseed ThreadRng in child processes:! rand::rng().reseed();L!,<OL Methods on `ThreadRng` are not reentrant-safe and thus should not be calledOL from an interrupt (e.g. a fork handler) unless it can be guaranteed that no@= other method on the same `ThreadRng` is currently executing./, [`ReseedingRng`]: crate::rngs::ReseedingRng#.\LrrȲP .   ,v.[t&&_ٟ_ ,'   &3џ_   O__ ddStorageddddZyџ!_   "N_"_ dd/ddddZyr5p-* An iterator over multiple slice elements.qq This struct is created byqZW [`IndexedRandom::choose_multiple`](trait.IndexedRandom.html#tymethod.choose_multiple).r r r87_77  77  7  77 6 7V7 $ 7$ 7 7 M 7>7 ] 77,777 w 777;7I777^77 4 7 4 7L 77 M 7M 7O 7E7Q 777{777 e 77 77 7L77ث7A7٥777 O 7U7Q7I777 [ 7e7 _ 77 x 7q7o7o7 o7 " 7n7l7 ) 7j7j7 7 7i73f7e7de7 7b7Nb7a7 "7w^7 ^7]7 79\7[7x[7 7W7V7 $7L$7R7Q7BQ7 47J7mJ7J7G7a<77737 =717-7,7)7 7$7 7!7 7 ,77 +7 >77 <77 E7 O7 7 O7W7 [7C7 b7g7!7 e7x7 g7M77H77>7=7 7T7 7$777)7c7h77T77v7 Z77 [77 ]7777777777Q7X7777`777I7+727ʖ7{77 7.71787M71747;7i7z777%77,7q777%7ݚ7,777I7777777 7 7!7U7"77#77$77%7)7&7^7'77(7ȏ7)77*7777ۅ77 77B779I7}7;I7|7=I7L{7?I7y7AI7w7CI7u7EI7s7GI7!q7II7n7KI7k7MI7h7OI7e7QI7R7R777Q7O7O7J7N7H7G7G7`7iF7pF7B7x)7A7@7<7})7;7^:7H/7-7-7Ϗ7,7,7+7(7,7'7%7"7,7!7877,7u777,777K7,7= 7d 77,7777,77,7v7,7g777,77787,7)7O7777777c7777777+77 77@ 7P7V777 7O 7 7K7Q77777[7"7(7n77y77_7E7K7H777W7ȸ777׶7H777W7ȳ777ױ7H777W7Ȯ777׬7H7777777ޜ7 77 7D7 7w7 77 7ݘ7 77 7C7 7v7 77 7ܔ7 77 7C77w7777G7777=77x7v7w7"J7u7\ 7u7\ 7At7\ 7ns7\ 7r7c 7q7k 7p7c 7"p7k 7Oo7c 7|n7k 7m7c 7l7k 7l7c 70k7k 7]j7c 7i7k 7h7c 7g7k 7g7c 7>f7k 7ke7c 7d7k 7W7W7:7U7U7:7O7 O77sN7M7M7L7?7>7>7 78>7>>7=75<7;<7P7;7:7:7;:7*07I/7O/77.7.7-7R"777J777747-7377`7777 `The Rust Rand Book4 5Rng::random_bool6rng() rand::rng()rngs::ThreadRngseq::IteratorRandom::chooseseq::IteratorRandomdistr::StandardUniformRng::random_rangeS 6 random()///Rng::random_ratioRng::random_iter5rand//S 6 Rng::fill Rng::randomA///5 ////S7"https://rust-random.github.io/book/8Adistr::Bernoulli6////5/55//4/8A5//6 /7/5 /5 7//4 85 //6////55# probability distribution6https://en.wikipedia.org/wiki/Probability_distributionDistribution[}https://crates.io/crates/statrs#  std::simd::mask32x4 Rng::sample00[[} statrs0/  std::simd::f32x4ׁ0/00V[[#https://crates.io/crates/rand_distr00std::num::NonZeroU8/ & std::simdς0ׁ0/0WV[ rand_distr00std::num҃0/0ς0 //5WV̈́00std::num::Wrapping0҃00? 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